EXITconstrained BICMID design using extended mapping
 Kisho Fukawa^{1}Email author,
 Soulisak Ormsub^{1},
 Antti Tölli^{2},
 Khoirul Anwar^{1} and
 Tad Matsumoto^{1, 2}
https://doi.org/10.1186/16871499201240
© Fukawa et al; licensee Springer. 2012
Received: 28 April 2011
Accepted: 9 February 2012
Published: 9 February 2012
Abstract
This article proposes a novel design framework, EXITconstrained binary switching algorithm (EBSA), for achieving near Shannon limit performance with single parity check and irregular repetition coded bit interleaved coded modulation and iterative detection with extended mapping (SIBICMIDEM). EBSA is composed of node degree allocation optimization using linear programming (LP) and labeling optimization based on adaptive binary switching algorithm jointly. This technique achieves exact matching between the Demapper (Dem) and decoder's extrinsic information transfer (EXIT) curves while the convergence tunnel opens until the desired mutual information (MI) point. Moreover, this article proposes a combined use of SIBICMIDEM with DopedACCumulator (DACC) and modulation doping (MD) to further improve the performance. In fact, the use of DACC and SIBICMID (noted as DSIBICMIDEM) enables the rightmost point of the EXIT curve of the combined demapper and DACC decoder (D_{dacc}), denoted as DemD_{dacc}, to reach a point very close to the (1.0, 1.0) MI point. Furthermore, MD provides us with additional degreeoffreedom in "bending" the shape of the demapper EXIT curve by choosing the mixing ratio of modulation formats, and hence the left most point of the demapper EXIT curve can flexibly be lifted up/pushed down with MD aided DSIBICMIDEM (referred to as MDSIBICMIDEM). Results of the simulations show that nearShannon limit performance can be achieved with the proposed technique; with a parameter set obtained by EBSA for MDSIBICMIDEM, the threshold signaltonoise power ratio (SNR) is only roughly 0.5 dB away from the Shannon limit, for which the required computational complexity per iteration is at the same order as a Turbo code with only memory2 convolutional constituent codes.
Keywords
1 Introduction
The discovery of Turbo code [1] in 1993 is a landmark event in the history of coding theory, since the code can achieve nearShannon limit performance. It is shown in [1] that the Turbo code composed of memory4 constituent convolutional codes can achieve 0.7 dB, in Signaltonoise power ratio (SNR), away from the Shannon limit. Various efforts have been made since then to achieve Turbo codelike performance without requiring heavy computational efforts for decoding.
Bitinterleaved coded modulation and iterative detection/decoding (BICMID) [2] has been recognized as a bandwidth efficient coded modulation scheme, of which transmitter is comprised of a concatenation of encoder and bittosymbol mapper separated by a bit interleaver. Iterative detectionanddecoding takes place at the receiver, where extrinsic log likelihood ratio (LLR), obtained as the result of the maximum a posteriori probability (MAP) algorithm for demapping/decoding, is forwarded to the decoder/demapper via deinterleaver/interleaver and used as the a priori LLR for decoding/demapping according to the standard turbo principle.
Performances of BICMID have to be evaluated by the convergence and asymptotic properties [3], which are represented by the threshold SNR and bit error rate (BER) floor, respectively. In principle, since BICMID is a serially concatenated system, analyzing its performances can rely on the area property [4] of the EXtrinsic Information Transfer (EXIT) chart. Therefore, the transmission link design based on BICMID falls into the issue of matching between the demapper and decoder EXIT curves.
Various efforts have been made seeking for better matching between the two curves to minimize the gap, while still keeping the tunnel open, aiming, without requiring heavy detection/decoding complexity, at achieving lower threshold SNR and BER floor. In [5], ten Brink et al. introduced a technique that makes good matching between the detector and decoder EXIT curves using low density parity check (LDPC) code in multiple input multiple output (MIMO) spatial multiplexing systems.
It has long been believed that for 4quadrature amplitude modulation (4QAM), the combination of Gray mapping and Turbo or LDPC codes achieves the optimal performance. However, Schreckenbach et al. [6] propose another approach towards achieving good matching between the two curves by introducing different mapping rules, such as nonGray mapping, which allows the use of even simpler codes to achieve BER pinchoff (corresponding to the threshold SNR) at an SNR value relatively close to the Shannon limit.
Another technique that can provide us with the design flexibility is extended mapping (EM) presented in [7, 8] where with 2^{ m }QAM, ℓ_{map} bits (ℓ_{map} > m), are allocated to one signal point in the constellation. With EM, the leftmost point of the demapper EXIT function has a lower value than that with the Gray mapping, but the rightmost point becomes higher. With this setting, the demapper EXIT function achieves better matching even with weaker codes such as short memory convolution codes as shown in [7]. However, there is a fundamental drawback with the structure shown in [7]; it still suffers from the BER floor simply because the demapper EXIT curve does not reach the topright (1.0,1.0) MI point.
In [9], Pfletschinger and Sanzi suggest that by using the memory1 rate1 recursive systematic convolutional code (RSCC), referred to as DACC located immediately after the interleaver, the error floor can be eliminated. Furthermore, it was shown by [10] that by replacing the RSCCcoded bits b_{ u }(P) with the accumulated bits b_{ c }(P) at every P bittimings, the technique of which is referred to as inner doping with doping ratio (1:P), the EXIT curve of DemD_{dacc} can be flexibly changed.
Several techniques have been proposed to determine optimal labeling pattern for the modulation (bit pattern vector allocated to each constellation point). The ideas of binary switching algorithm (BSA), which aims at labeling costs optimization, are presented in [6, 11]. However, the BSA based labeling optimization evaluates the labeling cost assuming that full a priori information is available. Hence, this approach only aims at lifting up as much the rightmost point of the demapper EXIT curve as possible. Yang et al. [12] introduce adaptive binary switching algorithm (ABSA) to obtain optimal labeling pattern, where optimality is defined by taking into account the labeling costs at multiple a priori MI points. Hence, ABSA changes the shape of the demapper EXIT curves more flexibly than BSA. However, the optimal labeling obtained in ABSA is on given codebasis since the code parameter optimization is not included in the ABSA iterations.
In our previous publication [13], we introduced a BICMID technique that uses even simpler codes, single parity check code (SPC) and irregular repetition code (IRC), combined with EM. For the notation simplicity, we refer our proposed BICMID structure in [13] to as SPCandIRC aided BICMID with EM (SIBICMIDEM). We investigated in [14] that linear programming (LP) technique can be applied for SIBICMIDEM to determine the optimal degree allocations for the IRC code with the aim of achieving desired convergence property. Moreover, in [15] we proposed a combined use of modulation doping (MD), originally proposed in [16, 17], which mixes the labeling rules for the extended nonGray mapping and the standard Gray mapping at a certain ratio. The technique proposed in [15] helps the leftmost point of the demapper slightly be lifted up to initiate the LLR exchange between the demapper and the decoder. This technique gives the additional degreeoffreedom in "bending" the shape of the demapper EXIT curve by choosing the mixing ratio and hence the leftmost point of the demapper EXIT curve can be flexibly lifted up/pushed down. This article proposes a combined use of SIBICMIDEM with DACC and MD. The DACC aided SIBICMIDEM is referred to as DSIBICMIDEM, and MD aided DSIBICMIDEM is referred to as MDSIBICMIDEM later on.
The primary goal of this article is to create a design framework for the optimization of SIBICMIDEM^{a} by combining those techniques into a unified iterative algorithm. To achieve the goal described above, this article proposes a new labeling pattern optimization technique, EXITconstrained Binary Switching Algorithm (EBSA). The gap between the two EXIT curves is taken into account in a repeatuntil loop that controls the EBSA algorithm. Hence, the process for determining the optimal degree allocation using LP [13, 14] is also included in the repeatuntil loop in EBSA.
The results of simulations show that nearShannon limit performance can be achieved with the proposed techniques; BER simulation results show that 4QAM EM with ℓ_{map} = 5, the threshold SNR is only roughly 0.5 dB away from the Shannon limit with MDSIBICMIDEM, for which required computational complexity for DemD_{dacc} is almost the same as a Turbo code with only memory2 convolutional constituency codes, per iteration.
This article is organized as follows; our proposed system structure is described in Section 2. Theoretical EXIT functions of the codes used in SIBICMIDEM are presented in Section 3. EBSA is introduced and detailed in Section 4, which is the core part of this contribution. In Section 5 numerical results are provided: in Section 5.1, convergence property of the proposed schemes described to confirm the effectiveness of EBSA; in Section 5.2, the results of BER performance evaluations are presented. In Section 6, computational complexity with the proposed technique is assessed briefly. Finally, we conclude this article in Section 7 with some concluding statements.
2 System model
2.1 Transmitter
The repetition times d_{ v }, referred to as variable node degree, may take different values in a block (transmission frame); if d_{ v }takes several different values in a block, such code is referred to as having irregular degree allocations. It is assumed throughout this article that d_{ c }takes only one identical value as in [5].
bits per symbol, where a_{ i }represents the ratio of variable nodes having degree ${d}_{{v}_{i}}$ in a block and M is the number of node degree allocations.
The coded bit sequence is bitinterleaved, and segmented into ℓ_{map}bit segments, and then each segment is mapped on to one of the 2^{ m }constellation points for modulation. The complexvalued signal modulated according to the mapping rule is finally transmitted to the wireless channel. Since ℓ_{map} > m with EM, more than one label having different bit patterns in the segment are mapped on to each constellation point. However, there are many possible combinations of the bit patterns, hence determining of the optimal labeling pattern plays the key role to achieve limitapproaching performance.
2.2 Channel
where $s=\left[{b}_{1}\phantom{\rule{0.3em}{0ex}}{b}_{2}\dots {b}_{{\ell}_{\text{map}}}\right]$ is labeling pattern and ψ(·) is the mapping function as indicated in Figure 1. n is zero mean complex AWGN component with variance ${\sigma}_{n}^{2}$ (i.e.,〈x(k)^{2}〉 = 1, 〈n(k)〉 = 0 and $\u27e8{\leftn\left(k\right)\right}^{2}\u27e9={\sigma}_{n}^{2}$) for ∀k.
2.3 Receiver
where S_{0} (S_{1}) indicates the set of the labeling pattern s having the v th bit being 0(1), and L_{a,Dem}(b_{ ρ }(s)) is the demapper's a priori LLR fed back from the decoder corresponding to the ρ th position in the labeling pattern s.
Decoding takes place segmentwise where, because of the irregular code structure, the variable node degrees ${d}_{{v}_{i}}$ have different values segmentbysegment. Structure of the decoder as well as decoding algorithm is detailed in the previous publications, e.g., in [13, 14, 20]. Therefore, only summary of the algorithm is provided in this article.
This process is performed for the other variable nodes in the same segment having the variable node degree ${d}_{{v}_{i}}$, and also for all the other segments independently in the same transmission block. Finally, the updated extrinsic LLRs obtained at the each variable node are interleaved, and fed back to the demapper. For the final bitwise decision, a posteriori LLR output from the decoder is used.
2.4 DSIBICMIDEM
Reference [20] proposes a combined use of DACC with SIBICMIDEM (DSIBICMIDEM).^{b} The purpose of introducing DACC is to lift the rightmost point of demapper EXIT curve up to reach the (1.0,1.0) MI point so that the BER floor with SIBICMIDEM can be eliminated. In this system structure, DACC is placed between the interleaver and mapper as shown in Figure 1 of [20]. The coded bit sequence is bitinterleaved, and input to the DACC with doping ratio of (1:P). To keep the DACC's code rate equal to one, the interleaver's output is replaced by a DACCcoded bit at every P th bit.
2.5 MDSIBICMIDEM
3 EXIT analysis
Since detailed investigation for the effect of EM on the shape and the demapper's EXIT function is provided in the previous publications, e.g., [20], they are not provided in this article. For those readers who are interested in this issue can refer [7, 13, 20].
Hence, it is found that the key of achieving the best matching between the demapper and the decoder EXIT curves is to jointly optimize the labeling pattern and the variable node degree distribution a_{ i }.
4 Framework for EBSAbased DSIBICMIDEM design
4.1 Optimal node degree allocation using LP
More details are given in Appendix 1. Furthermore, to find the optimal check node degree d_{ c }, this article proposes a bruteforce search (all possible value search),^{c} as summarized in Algorithm 1.
Algorithm 1 Optimal degree allocation algorithm
Initialize ${d}_{{v}_{i}}$ and a_{ i }values.
for d_{ c }= 2 to max d_{ c }do
Perform LP for Equation (14) with fixed d_{ c }and obtain optimal distribution a_{ i }for each ${d}_{{v}_{i}}$.
Calculate code rate R using d_{ c }, d_{ v }and a_{ i }.
end for
Find ${d}_{{c}_{\text{opt}}}$ and a_{opt} achieving R → max.
return ${d}_{{c}_{\text{opt}}}$ and a_{opt}.
4.2 EBSA framework
In [12], Yang et al. introduce the idea of Adaptive BSA (ABSA) which takes into account the costs at multiple a priori information points. The gap width between the demapper and the decoder EXIT curves is also taken into account, given the decoder EXIT curve. ABSA then obtains the optimal doping ratio in conjunction with determining the optimal labeling pattern. Hence, opening of the convergence tunnel until the (1.0, 1.0) MI point is guaranteed with this technique. However, ABSA does not change the code parameters in optimization process, and therefore, optimality is only on given codebasis.
Since both the ABSA and EBSA algorithms, in common, are based on the BSA, as well as the same cost definition, BSA and the cost are summarized in Appendices 2 and 3, respectively, for the completeness of the article. This article's proposed EBSA algorithm is summarized in Algorithm 2. It should be noticed that the processes for determining the optimal doping ratio, the d_{ c }value, and the LP based code parameter optimization are all included in a single repeatuntil loop. This indicates that the code parameters are also changed in the EBSA framework.
It should be further noticed that the horizontal and vertical gap widths evaluation, as descriptively summarized in Figure 5, is included in the repeatuntil loop. With the EBSA framework, the labeling pattern used in the LPbased degree allocation optimization for DSIBICMIDEM are obtained by lowering the cost of ${Z}_{{\ell}_{\text{map}}1}$ (at rightmost MI point corresponding to the case with full a priori information) as much as possible, while still keeping the vertical gap smaller than the predefined value δ_{ w }. Hence, other costs ${Z}_{0},\dots ,{Z}_{{\ell}_{\text{map}}2}$ are ignored in the LP based optimization.
5 Numerical results
5.1 Convergence property analysis
The EBSA optimization technique is a design framework for BICMID, and therefore applicable not only to MDSIBICMIDEM, but also to other structures, as described in Endnote "a" in Section 1. To demonstrate the performances superiority with the optimization techniques described in this article, EXIT curves were calculated for several designs described in the previous sections, aiming at the turbo cliff to happen at SNR = 0.8 dB and 3.1 dB, as examples.
5.1.1 SIBICMIDEM with node degree optimization using LP
Initial degree distributions
d _{ v }  1  2  3  4  ⋯  6  7  8  9  30 

a  $\frac{1}{30}$  $\frac{1}{30}$  $\frac{1}{30}$  $\frac{1}{30}$  ⋯  $\frac{1}{30}$  $\frac{1}{30}$  $\frac{1}{30}$  $\frac{1}{30}$  $\frac{1}{30}$ 
Optimized degree distributions
d _{ v }  1  2  3  4  5  6  ⋯  23  ⋯  30 

a  0  0  0  0  0.9438  0.0419  ⋯  0.0143  ⋯  0 
ϵ settings
w  1  ⋯  6  7  8  9  10  11  ⋯  24 

ϵ _{ w }  0.001  ⋯  0.001  0.001  0.001  0.001  0.001  0.001  ⋯  0 
Algorithm 2 EXITconstrained binary switching algorithm (EBSA)
$\lambda =\left[{\lambda}_{0}\phantom{\rule{0.3em}{0ex}}{\lambda}_{1}\cdots {\lambda}_{{\ell}_{\text{map}}2}\phantom{\rule{0.3em}{0ex}}{\lambda}_{{\ell}_{\text{map}}1}\right]=\left[0\phantom{\rule{0.3em}{0ex}}0\cdots 0\phantom{\rule{0.3em}{0ex}}1\right]$
Initialization: Generate labeling pattern s randomly as well as degree distribution d_{ v }empirically
for P = 2 and d_{ c }= 2 to max P and max d_{ c }, respectively do
repeat
Draw demapper EXIT curve based on the given labeling pattern s obtained as the result of the latest iteration
for i = 1 to N_{max}^{d} do
Conduct BSA with ${Z}_{{\ell}_{\text{map}}1}=\stackrel{\u0304}{Z}$ and ${Z}_{{\ell}_{\text{map}}1}^{h}={\stackrel{\u0304}{Z}}^{h}\left(h=0,\dots {2}^{{\ell}_{\text{map}}1}\right)$
end for
Draw the demapper's EXIT curve using the obtained labeling pattern.
Draw the decoder's EXIT curve using degree distribution using LP for Equation (14) with the obtained labeling pattern.
λ_{ q }= λ_{ q } 1
end if
Select the labeling pattern and decoder node distribution that has minimum gap.
until Gap between the demapper and decoder EXIT curve becomes smaller than the threshold for each MI points tested.
end for
Select the optimal parameter set that minimizes the gap
It is found by carefully looking at the rightmost part of the curves that the intersection point of the (ii)(iv) decoder curves and the demapper curve indicated by (*) in the figure are found to be slightly closer to the extrinsic MI = 1.0 than the empirically designed case. However, the rate of the code obtained by LP is slightly lower than the rate of the code with empirically obtained degree allocation, and the intersection point of the two EXIT curves is still quite apart from the (1.0,1.0) MI point. Therefore, LP alone can lower the BER floor, but cannot increase the spectrum efficiency in those cases.
5.1.2 DSIBICMIDEM with node degree optimization using LP technique
Similar result can be observed from Figure 8b, where optimization was performed for SNR = 3.1 dB. Both in Figures 8a,b, the two EXIT curves intersect at a point very close to the (1.0,1.0) MI point. Therefore, no BER floor (or, at least invisible in the BER value range shown in the figure) and higher spectrum efficiency compared to the empirically designed SIBICMIDEM are expected.
5.1.3 MDSIBICMIDEM with EBSA
δ settings
w  1  ⋯  6  7  8  9  10  11  ⋯  24 

δ _{ w }  0.001  ⋯  0.001  0.001  0.001  0.001  0.001  0.001  ⋯  0 
From Figure 10a, very close matching between the DemD_{dacc} and the decoder EXIT curves can be observed from the starting point to the end. Moreover, now the DemD_{dacc} EXIT curve starts from (0,0.0064) and thereby, LLR exchange can be initiated, and hence the trajectory can reach the target MI point close enough to the (1.0,1.0) point. Similar characteristics can be observed in Figure 10b, where the optimization was performed for SNR = 3.1 dB.
5.2 BER performances
6 Complexity assessment
With ℓ_{map} = 5, there are 32 labeling patterns in total, where each of the sets S_{0} and S_{1} contains 16 patterns. Hence, the probabilities for the 16 patterns have to be summed up when calculating the numerator and the denominator of (5). Since the BCJR algorithm requires forward and backward processing and each state emits two branches corresponding to the systematic input being 0 and 1, the computational complexity for the demapper having 16 labeling patterns both in the numerator and the denominator is equivalent to the decoding complexity of memory3 convolutional code using the BCJR algorithm $\left(3={\text{log}}_{2}\left(8\right)={\text{log}}_{2}\left(\frac{16}{2}\right)\right)$. Furthermore, since Turbo code requires at least two constituency codes [1], the complexity estimated above is also roughly equivalent to that required by a Turbo code having two memory2 constituent convolutional codes $\left(2={\text{log}}_{2}\left(4\right)={\text{log}}_{2}\left(\frac{8}{2}\right)\right)$.
Calculation cost of 4QAM EM
Addition/Subtraction  Multiplication  Division  

4QAM EM  ${2}^{{\ell}_{\text{map}}}{\ell}_{\text{map}}^{2}$  $2{\ell}_{\text{map}}^{2}2$  3ℓ_{map} 
7 Conclusions
This article has proposed a design framework, EBSA, and applied it to our proposed BICMID techniques, SIBICMIDEM, DSIBICMIDEM and MDSIBICMIDEM. Since EBSA takes into account the horizontal and vertical gap widths between the DemD_{dacc} and decoder's EXIT curves at the predefined several MI points, it can determine the optimal labeling pattern for EM and degree allocations simultaneously. In fact, when EBSA is applied to DSIBICMIDEM, two curves exactly match, and surprisingly the leftmost point of the EXIT curve of DemD_{dacc} is determined to be the (0.0, 0.0) MI point, and hence LLR exchange can not be initiated. To avoid this situation, this article introduced the MD technique, by which the leftmost point of the DemD_{dacc} EXIT curves can be lifted up slightly while the other part still exactly matched. As the result, very closeShannon limit performance can be achieved without requiring heavy computational burden with ℓ_{map} = 5 EM 4QAM; the complexity is almost the same level as a Turbo code with only memory2 constituency codes.
The following three issues have to be noted in concluding this article, since this special issue has two focal points, "Algorithm and Implementation Aspects":

The proposed EBSA is applicable to BICMID techniques using other codes, so far as the degree allocation optimization can be performed using LP. LDPCaided BICMID [23] and irregular convolutional codeaided BICMID [16] belong to this category. This is the reason why call EBSA "framework" rather than "technique".

The tradeoff between performance and complexity due to iterations can well be managed with EBSA by properly setting the horizontal and vertical gap parameters, ϵ and δ, respectively, at several MI points. Even relatively large gap parameters are used so that not too many iterations are required, still arbitrary low BER can be achieved because the two curves reach a point close enough to the (1.0, 1.0) MI point.

Application of the EBSA framework to higher order modulation is left as future study.
Appendix 1: Node degree optimization using LP
 1.
Code rate has to be lower than but as close to the capacity as possible.
 2.
Keep the convergence tunnel open between the demapper and decoder EXIT curve until the desired intersection point and the point should be as close to the (1.0, 1.0) MI point as possible.
 3.
Total of node degrees distributions has to be always 1.
Now, given the fact that the optimization parameter in (19) is only a_{ i }and the other terms are constant and furthermore, the index and constraints are both linear function of the optimization variable a_{ i }. Hence, the problem can be solved by using LP techniques.
Appendix 2: Summary of BSA
for $h=0,\phantom{\rule{0.3em}{0ex}}1,\dots ,{2}^{{\ell}_{\text{map}}}1$. The BSA is summarized in Algorithm 3.
Algorithm 3 Binary switching algorithm (BSA)
repeat
Initialization: generate labeling pattern randomly.
Select the symbol s_{ high }which has the highest cost ${Z}_{{\ell}_{\text{map}}1}^{h}$.
Find the symbol s_{ low }which can achieve maximum reduction of the total
cost ${Z}_{{\ell}_{\text{map}}1}$ by swapping the positions of s_{ high }and s_{ low }.
if s_{ low }exists. then
Swap s_{ high }and s_{ low }.
Update ${Z}_{{\ell}_{\text{map}}1}$ according to the new labeling pattern.
else
Set the symbol with the second highest cost as s_{ high }.
end if
until There is no pair of symbols to switch
A problem with this approach is, however, that the cost is calculated assuming the availability of a full a priori information and thus, ${Z}_{{\ell}_{\text{map}}1}$ affect only the right most point of the DemD_{dacc} EXIT curve, and it does not consider the matching between the demapper and decoder EXIT curves. This approach is reasonable only when the objective is to force the rightmost point of the demapper curve to reach as close to the (1.0,1.0) MI point as possible. However, this article has already proposed the use of DACC which already makes it possible for the demapper EXIT curve to reach a point close enough to the (1.0,1.0) MI point. In the subsection 4D, a novel algorithm, EBSA, is introduced to obtain a labeling pattern aiming at better matching between the two curves, assuming the use of DACC.
Appendix 3: Cost functions for ABSA and EBSA
Those costs affect the shape of the demapper EXIT curve. Figure 6 shows an intuitive example for ℓ_{map} = 5.
Endnotes
^{a}The proposed technique is also applicable to other BICMID techniques, so far as they use LP for degree allocation optimization. LDPC aided BICMID as well as irregular convolutional code aided BICMID belong to this class. This is the reason on why we call the technique proposed in this article "framework". ^{b}Combined use of DACC with BICMID itself was first proposed by [9]. The technique is also used in [12] to determine the optimal doping rate when it is combined with BSA. ^{ c }d_{ c }has indirect effect to the LP optimization, but to the code rate. ^{d}Schreckenbach et al. [6] show that N_{max} = 100 is enough. ^{e}The range is defined as: MI(Z_{ q }+ ΔZ_{q}) for q = 0, MI(Z_{ q }± ΔZ_{ q }) for 1 ≤ q ≤ ℓ_{map}  2 and MI(Z_{ q } ΔZ_{ q }) for q = ℓ_{map}  1, where Δ = 1/(ℓ_{map}  1).
Declarations
Acknowledgements
This research was in part supported by the Japanese government funding program, GrantinAid for Scientific Research (B) No. 20360168 and (C) No. 22560367, and in part by Finland distinguished professor program funded by Finnish National Technology Agency Tekes. The authors are highly thankful for valuable technical comments and suggestions given by Mr. Takehiko Kobayashi of Hitachi Kokusai Electric Inc. We also acknowledge Mr. Xin He of Information Theory and Signal Processing Lab., School of Information Science, JAIST for his valuable opinions and suggestions to improve the quality of this article.
Authors’ Affiliations
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